arXiv:1801.07428v1 [math.DG] 23 Jan 2018
FUNCTIONS IN SASAKIAN MANIFOLDS
∗SHU-CHENG CHANG, †YINGBO HAN, AND∗∗CHIEN LIN
Abstract. This is the very first paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K¨ahler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.
1. Introduction
S.-Y. Cheng and S.-T. Yau( [Y1], [CY]) derived the well-known gradient estimate for pos- itive harmonic functions and obtained the classical Liouville theorem, which states that any bounded harmonic function is constant in complete noncompactm-dimensional Riemannian manifolds with nonnegative Ricci curvature. Yau conjectured that the dimension hd(M) of the space Hd(M) consisting of harmonic functions of polynomial growth of degree at most d, is finite for each positive integer d and satisfies the estimate:
hd(M)≤hd(Rm).
1991Mathematics Subject Classification. Primary 32V05, 32V20; Secondary 53C56.
Key words and phrases. CR heat kernel, CR Yau’s uniformization conjecture, CR holomorphic function, Sharp dimension estimate.
∗Shu-Cheng Chang and∗∗Chien Lin are partially supported in part by the MOST of Taiwan.
†(Corresponding author) Yingbo Han is partially supported by an NSFC grant No. 11201400, Nanhu Scholars Program for Young Scholars of Xinyang Normal University and the Universities Young Teachers Program of Henan Province (2016GGJS-096).
1
Colding and Minicozzi ([CM]) affirmatively answered the first question and proved that
hd(M)≤C0dm−1
for manifolds of nonnegative Ricci curvature. Later, Li ([Li2]) produced a shorter proof requiring only the manifold to satisfy the volume doubling property and the mean value inequality. For the latter question, the sharp upper bound estimate is still missing except for the special cases m = 2 or d = 1 obtained by Li-Tam ([LT], [LT1]) and Kasue ([K]), and the rigidity part is only known for the special case d = 1 obtained by Li ([Li1]) and Cheeger-Colding-Minicozzi ([CCM]).
In [CKT], by modifying the arguments of [Y1], [CY], and [CKL], Chang, Kuo and Tie derived a sub-gradient estimate for positive pseudoharmonic functions in a complete non- compact pseudohermitian manifold (M, J, θ). This sub-gradient estimate can serve as the CR version of Yau’s gradient estimate. As an application of the sub-gradient estimate, the CR analogue of Liouville-type theorem holds for positive pseudoharmonic functions. In the recent paper ([CCHT]), we study the CR analogue of Yau’s conjecture on the space Hd(M) consisting of all pseudoharmonic functions of polynomial growth of degree at most d in a complete noncompact pseudohermitian manifold. We showed that the first part of CR Yau’s conjecture holds for pseudoharmonic functions of polynomial growth.
In K¨ahler geometry, Yau ([ScY]) proposed a variety of uniformization-type problems on complete K¨ahler manifolds with nonnegative holomorphic bisectional curvature. The first conjecture is
Conjecture 1. If M is a complete noncompact n-dimensional K¨ahler manifold with non- negative holomorphic bisectional curvature, then
dimC(Od(Mn))≤dimC(Od(Cn)).
The equality holds if and only if M is isometrically biholomorphic to Cn. Here Od(Mn) denotes the family of all holomorphic functions on a completen-dimensional K¨ahler manifold M of polynomial growth of degree at most d.
In [N], Ni established the validity of this conjecture by deriving the monotonicity formula for the heat equation under the extra assumption that M has maximal volume growth
r→lim+∞
V ol(Bp(r)) r2n ≥c
for a fixed point p and a positive constant c. Later, in [CFYZ], Chen, Fu, Yin, and Zhu improved Ni’s result without the assumption of maximal volume growth. One should refer to [Liu1] for more general results recently.
The second conjecture is
Conjecture 2. If M is a complete noncompactn-dimensional K¨ahler manifold with nonneg- ative holomorphic bisectional curvature, then the ring OP(M) of all holomorphic functions of polynomial growth is finitely generated.
This one was solved completely by G. Liu ([Liu2]) quite recently. He mainly deployed four techniques to attack this conjecture via Cheeger-Colding-Tian’s theory ([ChCo1], [ChCo2], [CCT]), methods of heat flow developed by Ni and Tam ([N], [NT1], [NT4]), Hormander L2-estimate of ∂ ([De]) and three circle theorem ([Liu1]).
The third uniformization conjecture is
Conjecture 3. If M is a complete noncompactn-dimensional K¨ahler manifold with positive holomorphic bisectional curvature, then M is biholomorphic to the standard n-dimensional complex space Cn.
The first giant progress pertaining to the third conjecture could be attributed to Mok, Siu and Yau. In their papers ([MSY] and [M1]), they showed that, under the assumptions of the
maximal volume growth and the scalar curvature R(x) decays as 0≤r(x)≤ C
(1 +d(x, x0))2+ǫ
for some positive constantCand any arbitrarily small positive numberǫ, a complete noncom- pact n-dimensional K¨ahler manifold M with nonnegative holomorphic bisectional curvature is isometrically biholomorphic to Cn. A Riemannian version was proved in [GW2] shortly afterwards. Since then there are several further works aiming to prove the optimal result and reader is referred to [M2], [CTZ], [CZ], [N2], [NT1], [NT2] and [NST]. For example, A. Chau and L. F. Tam ([CT]) proved that a complete noncompact K¨ahler manifold with bounded nonnegative bisectional curvature and maximal volume growth is biholomorphic to Cn. Recently, G. Liu ([Liu3]) confirmed Yau’s uniformization conjecture when M has maximal volume growth.
This is the very first paper to focus on the CR analogue of Yau’s uniformization con- jectures in a complete noncompact pseudohermitian (2n+ 1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K¨ahler geometry. We refer to the next section for the detailed notations. The following is the so-called CR Yau’s uniformization conjecture.
Conjecture 4. (CR Yau’s Uniformization Conjecture) Let M be a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with positive pseudohermitian bi- sectional curvature. Then M is CR biholomorphic to the standard Heisenberg group Hn = Cn×R.
In this paper, it is very natural to concerned the CR analogue of Yau’s first conjecture in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature. A smooth complex-valued function on a pseudohermitian (2n+ 1)-manifold (M, J, θ) is called CR-holomorphic if
∂bf = 0.
For any fixed pointx∈M, a CR-holomorphic functionf is called to be of polynomial growth if there are a nonnegative number d and a positive constant C =C(x, d, f), depending on x, d and f, such that
|f(y)| ≤C(1 +dcc(x, y))d
for all y ∈ M, where dcc(x, y) denotes the Carnot-Caratheodory distance between x and y. In the following sections, we sometimes would use the notation r(x, y) for the Carnot- Caratheodory distance. Furthermore, we could define the degree of a CR-holomorphic func- tion f of polynomial growth by
deg (f) = inf
d ≥0
|f(y)| ≤C(1 +dcc(x, y))d ∀y∈M, f or some d ≥0 and C =C(x, d, f)
as well as the aforementioned holomorphic case. In fact, the definition above is independent of the choice of the pointx∈M. Finally we denoteOdCR(M) the family of all CR-holomorphic functions f of polynomial growth of degree at most d with T f(x) = f0(x) = 0 :
OdCR(M) ={f(x)|∂bf(x) = 0, f0(x) = 0 and |f(x)| ≤C(1 +dcc(x, y))d for some constantC}. Now we explain the extra condition T f = 0 in the definition of OdCR(M) in a Sasakian manifold. Follow the notion as in [FOW] : Let{Uα}α∈Abe an open covering of the Sasakian manifold (M2n+1, g) and πα :Uα →Vα ⊂Cnsubmersion such thatπα◦π−β1 :πβ(Uα∩Uβ)→ πα(Uα∩Uβ) is biholomorphic. On each Vα, there is a canonical isomorphism
dπα :Dp →Tπα(p)Vα
for any p ∈ Uα, where D = kerθ ⊂ T M. Since T generates isometries, the restriction of the Sasakian metric g to D gives a well-defined Hermitian metric gαT onVα. This Hermitian metric in fact is K¨ahler. More precisely, letz1, z2,···, znbe the local holomorphic coordinates onVα. We pull back these to Uα and still write the same. Let xbe the coordinate along the
leaves withT = ∂x∂ .Then we have the local coordinate{x, z1, z2,· · ·, zn} onUα and (D⊗C) is spanned by the form
Zα = ∂
∂zα −θ ∂
∂zα
T
, α= 1,2,· · ·, n.
Since i(T)dθ = 0,
dθ(Zα, Zβ) =dθ( ∂
∂zα, ∂
∂zβ).
Then the K¨ahler 2-form ωTα of the Hermitian metric gαT on Vα, which is the same as the restriction of the Levi form 12dθ to Dfnα, the slice {x = constant} in Uα, is closed. The collection of K¨ahler metrics {gTα} on{Vα} is so-called a transverse K¨ahler metric. We often refer to 12dθ as the K¨ahler form of the transverse K¨ahler metric gTin the leaf space Dfn. As an example,
Zα = ∂z∂α +izα ∂∂t nα=1 is exactly a local frame on in the (2n+ 1)-dimensional Heisenberg group Hn = Cn×R. Here
θ =dt+iX
α∈In
(zαdzα−zαdzα)
is a pseudohermitian contact structure on Hn and T = ∂t∂. In this case, Dfn=Cn and then (1.1) dimCOd(Cn) = dimC OCRd (Hn)
.
Now, for a nontrivial functionf ∈ OCRd (M) withf(x) = 0, we would define the vanishing order of f at x∈M by
ordx(f) = max{m∈N | Dαf = 0, ∀ |α|< m }, where Dα = Y
j∈In
Zjαj with α= (α1, α2,..., αn).
Now we state our main theorem in this paper.
Theorem 1.1. If (M, J, θ)is a complete noncompact pseudohermitian (2n+ 1)-manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature, then
(1.2) dimC OdCR(M)
≤dimC OCRd (Hn)
for each d∈N and Hn = Cn×R is the (2n+ 1)-dimensional Heisenberg group.
Remark 1.1. 1. In order to prove Theorem 1.1, it follows from (1.1) that it suffices to show that
(1.3) dimC OCRd (M)
≤dimCOd(Cn)
in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.
2. It is interesting to know whether M is CR equivalent to the Heisenberg group Hn if the equality holds
dimC OdCR(M)
= dimC OdCR(Hn) .
The method here is inspired mainly from [N] and [CFYZ] which is organized as follows.
In Section 2, we introduce some basic notions about pseudohermitian manifolds and the necessary results for this paper. In Section 3, we show the existence of solutions to the CR heat equation with the initial condition under some appropriate assumptions. In Section 4, we prove the CR analogue of the rough dimension estimate. In Section 5, we show the CR sharp dimension estimate. In Appendix, we would give the complete proof of the Lp- submean value inequality which is a key step in showing the existence of solutions to the CR heat equation in Section 3.
2. Preliminaries
We introduce some basic materials about a pseudohermitian manifold (see [L] and [DT]
for more details). Let (M, ξ) = (M, J, θ) be a (2n + 1)-dimensional, orientable, contact manifold with the contact structureξ. A CR structure compatible withξis an endomorphism J :ξ →ξ such that J2 =−1. We also assume that J satisfies the integrability condition: If XandY are inξ, then so are [JX, Y]+[X, JY] andJ([JX, Y]+[X, JY]) = [JX, JY]−[X, Y].
Let {T, Zα, Zα¯}α∈In be a frame of T M ⊗C, where Zα is any local frame of T1,0M, Zα¯ = Zα ∈ T0,1M ,and T is the characteristic vector field and In = {1,2, ..., n}.Then {θ, θα, θα¯}, the coframe dual to {T, Zα, Zα¯}, satisfies
dθ=ihαβθα∧θβ
for some positive definite hermitian matrix of functions (hαβ¯). If we have this contact structure, we call such M a pseudohermitian (2n+ 1)-manifold or strictly pseudoconvex CR (2n+ 1)-manifold as well.
The Levi formh , iLθ is the Hermitian form on T1,0M defined by hZ, WiLθ =−i
dθ, Z∧W .
We can extend h , iLθ to T0,1M by defining Z, W
Lθ = hZ, WiLθ for all Z, W ∈ T1,0M. The Levi form induces naturally a Hermitian form on the dual bundle of T1,0M, denoted by h , iL∗θ, and hence on all the induced tensor bundles. Integrating the Hermitian form (when acting on sections) over M with respect to the volume formdµ=θ∧(dθ)n, we get an inner product on the space of sections of each tensor bundle.
The pseudohermitian connection of (J, θ) is the connection ∇ onT M ⊗C(and extended to tensors) given in terms of a local frame Zα∈T1,0M by
∇Zα =ωαβ
⊗Zβ, ∇Zα¯ =ωα¯ β¯
⊗Zβ¯, ∇T = 0, where ωαβ are the 1-forms uniquely determined by the following equations :
dθα+ωβα∧θβ =θ∧τα τα∧θα = 0 ωβα+ωαβ = 0
.
We can write (by Cartan lemma) τα =Aαγθγ with Aαγ =Aγα. The curvature of Tanaka- Webster connection, expressed in terms of the coframe {θ =θ0, θα, θα¯}, is
Παβ = Παβ =dωαβ +ωαγ ∧ωβγ Πα0 = Π0α = Πα0 = Π0α = Π00 = 0
.
Webster showed that Πβα can be written
Ωαβ = Παβ +iτα∧θβ−iθα∧τβ =Rβ γδα θγ∧θδ+Wβγαθγ∧θ−Wβγαθγ∧θ where the coefficients satisfy
Rβαρ¯¯ σ =Rαβσ¯ ρ¯=Rαβ¯ σρ¯ =Rρ¯αβσ¯, Wβαγ¯ =Wγ¯αβ. Here Rβ γδα is the pseudohermitian curvature tensor, Rαβ¯ = Rγγ
αβ¯ is the pseudohermi- tian Ricci curvature tensor and Aαβ is the pseudohermitian torsion. R = hαβRαβ¯ denotes the pseudohermitian scalar curvature. Besides, we define the pseudohermitian bisectional curvature
Rα¯αββ(X, Y) =Rα¯αββXαXαYβYβ¯, the bitorsion tensor
Tαβ(X, Y) = 1
i(AαγXγYβ −AβγXγYα), the torsion tensor
T or(X, Y) =tr Tαβ
= 1 i
AαβXβYα−AαβXβYα ,
and the tensor
(divA)2(X, Y) =Aαγ,γAβδ,δXαYβ where X =XαZα, Y =YαZα inT1,0M.
We will denote the components of the covariant derivatives with indices preceded by comma. The indices{0, α,α¯}indicate derivatives with respect to{T, Zα, Zα¯}. For derivatives of a real-valued/complex-valued function, we will often omit the comma, for instance, uα =
Zαu, uαβ¯ = Zβ¯Zαu−ωαγ(Zβ¯)Zγu. The subgradient ∇bϕ of a smooth real-valued function ϕ is defined by
h∇bϕ, ZiLθ =Zϕ
forZ ∈Γ (ξ) where Γ (ξ) denotes the family of all smooth vector fields tangent to the conact plane ξ. We could locally write the subgradient ∇bϕ as
∇bu=uα¯Zα+uαZα¯.
Accordingly, we could define the subhessian Hessb as the complex linear map Hessb :T1,0M ⊕T0,1M −→T1,0M⊕T0,1M
by
(Hessbϕ)Z =∇Z∇bϕ for Z ∈Γ (ξ) and a smooth real-valued function ϕ.
Next we recall the following commutation relations. (see [L]) Letϕbe a smooth real-valued function, σ=σαθα be a (1,0)-form and ϕ0 =T ϕ,then we have
ϕαβ = ϕβα, ϕαβ¯−ϕβα¯ = ihαβϕ0
ϕ0α−ϕα0 = Aαβϕβ
σα,βγ −σα,γβ = i(Aαγσβ −Aαβσγ) σα,βγ −σα,γβ = −i
hαγAδβσδ−hαβAδγσδ
σα,βγ −σα,γβ = ihβγσα,0+Rα βγδ σδ
σα,0β −σα,β0 = Aγβσα,¯γ−Aαβ,¯γσγ σα,0 ¯β −σα,β0¯ = Aγβ¯σα,γ +Aγ¯β,α¯ σγ
.
Last, we would introduce the concept of the adapted metric. A family of Webster adapted metrics hǫ of a strictly pseudoconvex CR (2n+ 1)-manifold (M, J, θ) are the Riemannian
metrics
(2.1) hǫ =h+ 1
ǫ2θ2
where h=h, iLθ is the Levi metric. In [CC], they show that
∆ǫg = 2∆bg+ǫ2T2g for any real-valued smooth function g and
dµǫ = 1 ǫ2nn!dµ
where dµ(y) = dy = θ ∧(dθ)n is a volume form on (M, J, θ). Here ∆ǫ and dµǫ are the Riemannian Laplace operator and the Riemannian volume element with respective to the adapted Riemannian metric hǫ, respectively. In this paper, we choose ǫ = 2n−11n!such that dµǫ = 12dµ.
3. The CR heat equation
In this section, we will derive the essential fact about the existence of solutions to the CR heat equation on a complete noncompact pseudohermitian (2n+ 1)-manifold. First of all, we give the following lemmas so that one can establish the existence of solutions to the CR heat equation. It is also very useful to study the CR Poincar´e-Lelong equation ([CCHL]).
We will use the semigroup method in [BBGM]. It is known that the heat semigroup (Pt)t≥0 is given by
Pt= Z ∞
0
e−λtdEλ
for the spectral decomposition of ∆b =−R∞
0 λdEλ in L2(M). It is a one-parameter family of bounded operators on L2(M). We denote
Ptf(x) = Z
M
H(x, y, t)f(y)dµ(y).
Here H(x, y, t) > 0 is the so-called symmetric heat kernel associated to Pt. Due to hy- poellipticity of ∆b, the function (x, t) → Ptf(x) is smooth on M ×(0,∞), f ∈ C0∞(M).
Moreover
u(x, t) =Ptf(x) is a solution of CR heat equation
∂u
∂t = ∆bu u(x,0) =f(x)
.
Under these settings, we’re going to prove some results derived by estimating the CR heat kernelH(x, y, t). In the following,Vx(r) denotes the volume of the geodesic ball Bx(r) with respect to the Carnot-Carath´eodory distance r(x, y) between x and y, and r(x) = r(x, o) where o∈M is a fixed point.
Theorem 3.1. Let (M, J, θ) be a complete noncompact pseudohermitian (2n+ 1)-manifold with nonnegative pseudohermitian Ricci curvature tensors and vanishing pseudohermitian torsion. If f is a continuous function on M such that
1 V(Bo(r))
Z
Bo(r)
|f|(x)dx≤exp(ar2 +b)
for some positive constants a >0 and b >0, then the following initial value problem
(3.1)
(∂t∂ − △b)v(x, t) = 0 v(x,0) =f(x) has a solution on M ×(0,16aC ]; moreover,
v(x, t) = Z
M
H(x, y, t)f(y)dy,
where H(x, y, t) is the Heat kernel on (M, J, θ).
The proof is based on the effect estimate on the CR heat kernel, CR volume doubling property and Lp submean value inequality which do not need the stronger assumption on vanishing pseudohermitian torsion as in Proposition 3.1. However, for the later purpose as in the following sections, we do need the condition of vanishing torsion, in which the heat kernel estimate is obtained also in [BBGM]. Then here we focus on the case of a complete noncompact pseudohermitian (2n+ 1)-manifold of vanishing torsion only.
Proposition 3.1. ([CCHT],[BBGM]) Let(M, J, θ)be a complete pseudohermitian(2n+ 1)- manifold with the pseudohermitian Ricci curvature
Ric(X, X)≥k0hX, XiLθ
and
sup
i,j∈In
|Aij| ≤k1 <∞ and sup
i,j∈In
|Aij,¯ı|2 ≤k2 <∞, for X =XαZα ∈T1,0M and k0, k1, k2 are constants with k1, k2 ≥0. Then
(i) There exist positive constants C3, C4, C5 such that forx, y ∈M, t >0
(3.2) H(x, y, t)≤ C3
Vx
√t12 Vy
√t12 exp
−C5
d2cc(x, y)
t +C4κt .
(ii) There exist positive constants C6, C7, C8 such that for x, y∈M, t >0 (3.3) H(x, y, t)≥ C6
Vx
√texph
−C7dcc2(x, y)
t −C8κ(t+d2cc(x, y))i .
(iii) There exist positive constants C9, C10 such that for 0< s < t
(3.4) H(x, x, s)
H(x, x, t) ≤ t s
C9
eC10κ(t−s).
(iv) (CR Volume Doubling Property) For any σ > 1, then there exist a positive constant C1 such that
Vx(σρ)≤C1σ2C9e(C1σ2+C8)κρ2Vx(ρ). Here κ=κ(k0, k1, k2)≥0.
Lemma 3.1. Let (M, J, θ) be a complete noncompact pseudohermitian (2n+ 1)-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curvature tensors. Assume that u is defined by
u(x, t) = Z
M
H(x, y, t)f(y)dy,
on M ×[0, T] for some T >0. Here f is a nonnegative function with
(3.5) lim
r→+∞exp(−C(n)r2(x)
4t )
Z
Bo(r)
f(x)dx = 0.
Then for any 0< t≤r2, and p≥1, (3.6)
1 V(Bo(r))
R
Bo(r)up(x, t)dx ≤ C(n)[V(B1
o(4r))
R
Bo(4r)fp(x)dx +(1tR∞
4r exp(−C(n)s8t 2)s(V(B1o(s))R
Bo(s)f(x)dx)ds)p].
Proof. For anyp≥1 and r≥√ t,
(3.7)
R
Bo(r)up(x, t)dx = R
Bo(r)(R
MH(x, y, t)f(y)dy)pdx
≤ 2p−1[R
Bo(r)(R
Bo(4r)H(x, y, t)f(y)dy)pdx +R
Bo(r)(R
M\Bo(4r)H(x, y, t)f(y)dy)pdx].
We first will estimate the second term on the right hand side in (3.7). Now for x ∈ Bo(r) and y∈ M\Bo(4r), we have r(x, y) ≥ 34r(y). By the estimates of the CR heat kernel as in Proposition 3.1, we have
(3.8)
R
M\Bo(4r)H(x, y, t)f(y)dy
≤ C1R
M\Bo(4r) 1 V(Bx(√
t))exp(−C2r22t(x,y))f(y)dy
≤ C1R
M\Bo(4r) 1 V(Bx(r+√
t))(r+√√tt)2C9exp(−C2r22t(x,y))f(y)dy
≤ V(BCo1(√t))R
M\Bo(4r)(r+√√tt)2C9exp(−C2r22t(x,y))f(y)dy
≤ V(BCo1(√t))(r+√√tt)2C9R∞
4r[exp(−C4t2s2)(R
∂Bo(s)f)]ds
≤ V(BCo1(√t))(r+√√t
t )2C9R∞
4r[exp(−C4t2s2)(R
Bo(s)f)]d(st2)
≤ C1(r+√√tt)2C9R∞
4r[VV(B(Bo(s))
o(√
t))exp(−C24ts2)(V(B1
o(s))
R
Bo(s)f)]d(st2)
≤ C1R∞
4r(√st)4C9exp(−C4t2s2)(V(B1o(s))R
Bo(s)f)d(st2)
≤ C1tR∞
4r exp(−C28ts2)s(V(B1o(s))R
Bo(s)f)ds.
whereC9is the constant as in Proposition 3.1. Here, besides utilizing the inequalityr(x, y)≥
3
4r(y), we have used the volume doubling property and the assumption (3.5) when performing integration by parts in the fifth inequality. Note that when separating M\Bo(4r) into the shells, the concept of the Legendrian normal has come into our case (see [CCW]). From now on, once we deal with such process of the integration, we keep the idea in mind.
As for the first term on the right hand side in (3.7), by H¨older’s inequality and the fact ([BBGM]) that
Z
M
H(x, y, t)dy= 1, we have
( Z
Bo(4r)
H(x, y, t)f(y)dy)p ≤ Z
Bo(4r)
H(x, y, t)fp(y)dy.
Hence
(3.9)
R
Bo(r)(R
Bo(4r)H(x, y, t))f(y)dy)pdx ≤ R
Bo(r)
R
Bo(4r)H(x, y, t)fp(y)dydx
≤ R
Bo(4r)fp(y)(R
Bo(r)H(x, y, t)dx)dy
≤ R
Bo(4r)fp(y)dy.
From (3.7)-(3.9), we know that the Lemma holds.
From the preceding lemma, we could control the upper bound of subsolutions to the power ofpof the CR heat equation by its Lp-norm under the same hypotheses. On the other hand, on the course of proving the existence of solutions to the CR heat equation, we also need the Lp submean value inequality for p∈(0,∞), which will be shown in the Appendix.
Proposition 3.2. (CRLp submean value inequality ) Let(M, J, θ)be a complete pseudoher- mitian(2n+1)-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curva- ture tensors. Let Q(τ, x, r, s) = (s−τ r2, s)×Bx(r) andQδ(τ, x, r, s) = (s−δτ r2, s)×Bx(δr) for τ > 0, x ∈ M, r > 0, s ∈ R, δ ∈ (0,1). If τ > 0 and p ∈ (0,+∞) are given, then there exists a constant A(τ, ν, p)> 0 such that if u is a positive subsolution to the CR heat
equation (∂t∂ − △b)u≤0 in Q(τ, x, r, s), then, for 0< δ < δ′ ≤1,
sup
Qδ
{up} ≤ A(τ, ν, p)
(δ′−δ)2+νr2V (Bx(r)) Z
Qδ′
updµ,
where dµ=dµdt. Here the constant ν is the exponential constant in CR Sobolev inequality ([CCHT])
(3.10) ( Z
Bx(ρ)|ϕ|ν−22ν dµ)
ν−2ν
≤Csρ2V (Bx(r))−ν2 [ Z
Bx(ρ)|∇bϕ|2dµ+ρ−2 Z
Bx(ρ)
ϕ2dµ]
for any ϕ∈Cc∞(Bx(ρ)), x∈M.
Now we are ready to prove Theorem 3.1 :
Proof. For all j ≥1, let 0≤ϕj ≤1 be a smooth cut-off function such that ϕj = 1 on Bo(j) and ϕj = 0 on M\Bo(2j). Let fj = ϕjf be continuous with compact support. Hence one can solve (3.1) with the initial valuefj for all time. The solutionvj is given by
vj(x, t) = Z
M
H(x, y, t)fj(y)dy
for (x, t)∈M×(0,∞). The existence and uniqueness of such vj of the form could be found in [Li]. By Lemma 3.1, for 0< t≤min
r2,16aC , we have
1 V(Bo(r))
R
Bo(r)|vj|dx ≤ V(B1o(r))
R
Bo(r)
R
MH(x, y, t)|fj(y)|dydx
≤ C[V(B1
o(4r))
R
Bo(4r)|fj(y)|dy +1t R∞
4r exp(−Cs8t2)s(V(B1
o(s))
R
Bo(s)fj)ds]
≤ Ceb[exp(16ar2) +R∞
4r exp(−Cs16t2)d(st2)]
≤ Ceb[exp(16ar2) + 1].
Assume 16aC ≤R2. Let (τ, x, r, s, p, δ, δ′) = (2, o, R,16aC ,1,12,1) in the CRLp submean value inequality, and we know that |vj| is a subsolution to the CR heat equation, then we have
sup
Bo(R2)×(0,16aC )
|vj| ≤ (12)2+νA(2,ν,1)R2V(Bo(R)) Z
Q1
|vj|dµ
≤ 23+νA(2, ν,1)V(B1o(R)) Z
Bo(R)
|vj|dx
≤ Cexp(16aR2+b).
From this, it’s easy to see that, after passing to a subsequence, {vj}j∈Ntogether with their derivatives uniformly converge on compact sets on M ×(0,16aC ) to a solution v of the CR heat equation. Moreover, for any (x, t)∈ M×(0,16aC ) as in (3.8), we have
Z
M
H(x, y, t)dy−vj(x, t) ≤
Z
M
H(x, y, t)(f(y)−fj(y))dy
≤ Z
M\Bo(j)
H(x, y, t)|f(y)|dy
≤ CR∞
j exp(−Cs16t2)d(st2)
≤ CR∞
j2 t
exp(−Cτ16)dτ.
Let j tend to +∞, we have
v(x, t) = Z
M
H(x, y, t)f(y)dy.
This completes the proof.
4. Rough Dimension Estimates
After settling the existence of solutions to the CR heat equation, we will give the as- ymptotic behavior of the solutions. It’s a crucial step on the course of proving the sharp dimension estimate. And it’s worth to note that because of the following lemma, we could
drop the assumption thatM is of maximum volume growth in the hypotheses of the sharp di- mension estimate. That’s why the statement about the sharp dimension estimate in [CFYZ]
could hold without the assumption of the maximum volume growth as in [N].
Lemma 4.1. Let (M, J, θ) be a complete pseudohermitian (2n+ 1)-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curvature tensors. Assume that v(x, t) is a solution to the CR heat equation
(∂
∂t −∆b)v(x, t) = 0 on M ×(0,+∞) with the initial condition
v(x,0) = 2 log|f(x)|, where f ∈ OdCR(M). Then
(4.1) lim sup
t→+∞
v(x, t) logt ≤d.
Proof. From the fact f ∈ OdCR(M), we see that for all ǫ > 0, there’s a constant C′ = C′(f, x, d, ǫ)>0 such that
|f(y)| ≤C′(1 +r(x, y)d+ǫ).
Hereafter, in this proof, we would denote the positive constants byC′ =C′(f, x, d, ǫ) and C =C(n, d, ǫ). These constants may be different line by line.
Because
1 V(Bo(r))
Z
Bo(r)
log|f(x)|dx ≤exp(ar2+b)
for any a >0, then, from Theorem 3.1, we know the solutionv(x, t) has the form v(x, t) = 2
Z
M
H(x, y, t) log|f(y)|dy.
Hence
v(x, t) = 2 Z
M
H(x, y, t) log|f(y)|dy
= 2[
Z [r(x,y)≤√
t]
H(x, y, t) log|f(y)|dy+ Z [r(x,y)>√t]
H(x, y, t) log|f(y)|dy]
≤ Z [r(x,y)≤√
t]
H(x, y, t) ((d+ǫ) logt+C′)dy
+ Z [r(x,y)>√t]
H(x, y, t) (2 (d+ǫ) logr(x, y) +C′)dy
fort >1. By the CR heat kernel estimate and the CR volume doubling property in [CCHT]
and [BBGM], we obtain v(x, t)
≤ (d+ǫ) logt+C′+ Z [r(x,y)>√t]
(d+ǫ)H(x, y, t) log(r(x,y)t 2)dy
≤ C Z [r(x,y)>√t]
d+ǫ
V(Bx(√t))exp(−Cr(x,y)2t 2) log(r(x,y)t 2)dy+ (d+ǫ) logt+C′
≤ CX
k≥0
Z
[2k√t<r(x,y)≤2k+1√ t]
d+ǫ
V(Bx(√t))exp(−Cr(x,y)2t 2) log(r(x,y)t 2)dy+ (d+ǫ) logt+C′
≤ C(d+ǫ)X
k≥0
22(k+1)C9exp −22k−1C
2 (k+ 1)
+ (d+ǫ) logt+C′
≤ (d+ǫ) logt+C′+C.
Here the CR volume doubling property is used in the fourth inequality. Then we are
done.
Now we give another lemma which is also essential for the proof of the rough dimension estimate. In the proof of the following lemma, we would find that it’s closely related to the CR moment-type estimate as in [CF].
Lemma 4.2. Let (M, J, θ) be a complete pseudohermitian (2n+ 1)-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curvature tensors. Assume that v(x, t) is a
solution to the CR heat equation
(4.2)
∂
∂t −∆b
v(x, t) = 0
on M ×(0,+∞) with the initial condition
(4.3) v(x,0) = 2 log|f(x)|
for f ∈ OCRd (M), and
w(x, t) = ∂
∂tv(x, t). Then there exists a positive constant C such that
(4.4) C(n)ordxf ≤ lim
t→0+tw(x, t).
Proof. On account of the equation (4.2) and the initial condition (4.3), we have
w(x, t) = 2 Z
M
H(x, y, t)∆blog|f(y)|dy.
by the uniqueness theorem in [Do]. So by the CR heat kernel estimate in [CCHT], we know
(4.5)
w(x, t) = 2 Z
M
H(x, y, t)∆blog|f(y)|dy
≥ V(BxC(√t)) Z
M
exp(−Cd(x,y)t 2)∆blog|f(y)|dy
≥ V(BxC(√t)) Z [0,√t]
exp
−Crt2 (
Z
∂Bx(r)
∆blog|f(s)|dσ(s))dr
≥ V(BxC(√t)) Z
Bx(√t)
∆blog|f(y)|dy.
Furthermore, from the local property of the Sasakian manifolds and T f = 0, we have, as t <<1,
(4.6)
C V(Bx(√t))
Z
Bx(√t)
∆blog|f(y)|dy
= V C
(Bx(√t)) R√t
−√ t
R
Bxe(√t−l2) ∆yelog|f((ey, t))|dydle
≥ C
t2n+12 ordxe(f)R√t
−√ t
V(Bex(√t−l2))
(t−l2) dl by (4.7)
= C
t2n+12 ordx(f)R√t
−√ t
V(Bex(√t−l2))
(t−l2) dl
≥ C
t2n+12 ordx(f)R√t
−√
t(t−l2)n−1dl
= C
t2n+12 tn−12ordx(f)
= Ctordx(f).
Here we have used the identity
(4.7) ordex(f) = 1
2n lim
r→0+
r2 V (Bex(r))
Z
Bex(r)
∆ loge |f(y)e|dey
where ∆ denotes the Laplace operator on the slicee Dfn with the trasversal K¨ahler structure.
Also we adopt the adapted metric and
ordx(f) =ordxe(f),
which comes from the fact Zαf = ∂z∂fα, f0 = 0 and utilize the equality
(4.8)
Z s
−s
s2−l2n−1
dl=s2n−1√
π Γ (n) Γ(n+12)
in the third equality of (4.6) with s =√
t which is derived as follows : Rs
−s(s2−l2)n−1dl = s2n−1R1
−1(1−x2)n−1dx
= 2s2n−1R1
0 (1−x2)n−1dx
= s2n−1R1
0 (1−z)n−1 1√zdz
= s2n−1B(12, n)
= s2n−1 Γ(12)Γ(n)
Γ(n+12)
= s2n−1√π Γ(n)
Γ(n+12),
where B(z, w) and Γ (z) denote the beta function and the gamma function respectively.
This completes the proof of Lemma 4.2.
Prior to showing the sharp dimension estimate, we would obtain the dimension estimates of the rough version as follows :
Theorem 4.1. Let (M, J, θ) be a complete noncompact pseudohermitian (2n+ 1)-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curvature tensors. Then there exists a positive constant C(n) such that
(4.9) dimC OdCR(M)
≤C(n)dn.
Proof. We adopt the notations in the proof of Lemma 4.2. Set w(x, t) = ∂t∂v(x, t). By the similar deductions in [CF, Lemma 4.2.], replaced the moment-type estimate in [CF, (3.10) and (5.12)] by (4.1), we came out with [CF, (5.14)] and then
(4.10) ∂
∂t (tw(x, t))≥0.
In fact, we could check the hypotheses of Theorem 4.1 in [CF] from the upper bound of vαβ(x, t) by Cdlogt as in Lemma 4.1.
We first show that
(4.11) ordxf ≤Cd.
Basically, if we could prove
(4.12) tw(x, t)≤Cd
for t > 1, then we have (4.11) by Lemma 4.2. We observe that on account of the definition of OdCR(M) for all x∈M, there exists a constant Ce=Ce(f, x, d, ǫ)>0 such that
log+|f(y)| ≤Ce+d(log (1 +d(x, y))). Accordingly we have
(4.13)
v(x, t)
≤ V(BxC(√t)) R+∞
0 [exp
−Cr2t2 (
Z
∂Bx(r)
log+|f(s)|dσ(s))]dr
≤ V(BxC(√t)) R√+∞
t [exp
−Cr2t2 (
Z
∂Bx(r)
log+|f(s)|dσ(s))]dr+Cdlog (1 +t)
≤ V(BxC(√t)) R√+∞
t [exp
−Cr2t2
Cr t (
Z
Bx(r)
log+|f(y)|dy)]dr+Cdlog (1 +t)
≤ CR√+∞ t [
√r t
2C
exp
−Cr2t2
Cr
t dlog (1 +r)]dr+Cdlog (1 +t)
≤ CR√+∞
t [tC+1τC exp −C2tτ
dlog (1 +τ)]dτ +Cdlog (1 +t)
≤ CR+∞ 0
sCexp (−s)sdlog (1 +t)
ds+Cdlog (1 +t)
≤ Cdlog (1 +t) +Cdlog (1 +t) Γ (C+ 2)
≤ Cdlogt
for t > 1. From (4.13), we claim that the inequality (4.12) holds; for if there’s some small positive constant ǫ and t0 >1 such that
tw(x, t)>(C+ǫ)d
for t > t0. Here we have utilized the monotonicity of tw(x, t). Therefore, by integrating both sides, we have
v(x, t)≥(C+ǫ)dlogt−A
where the constant A is independent of t. But this contradicts the inequality (4.13). So we obtain (4.12). As a result, we deduce (4.11).
Now we fix the constant C chosen in (4.11) and may assume such constant is a positive integer. Next we want to settle the dimension estimate (4.9). In spite of the proof of (4.9) is the same as the one in [M1], we would write it down for completeness. Let k(n) be a constant satisfying
q(m) =
n+m n
< k(n)mn. Consider the map
Φ :OCRd (M)−→Cq(C)
f 7−→(Dαf)|α|≤Cd. From (4.11), we see that Φ is injective. Suppose
dimC OdCR(M)
> C′dn
where C′ is chosen withC′ > k(n)Cn, this implies that dimC OdCR(M)
> k(n) (Cd)n> q(C).
However, this contradicts with the fact that Φ is injective. We complete the proof.
5. Sharp Dimension Estimates
Subsequently, we give another lemma also substantial to the proof of the sharp dimension estimate. In contrast with Lemma 4.1, one could find the following lemma is the stronger version of that. In fact, that is why we could derive the more advanced result-the sharp dimension estimate than the one in the last section.
Lemma 5.1. let (M, J, θ) be a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature. Assume that
v(x, t) is a solution to the CR heat equation ∂
∂t −∆b
v(x, t) = 0 on M ×(0,+∞) with the initial condition
v(x,0) = 2 log|f(x)| for f ∈ OCRd (M), and
w(x, t) = ∂
∂tv(x, t). Then
(5.1) lim
t→0+tw(x, t) =ordxf.
Proof. From the definition ofw(x, t), we have
(5.2) w(x, t) = 2
Z
M
H(x, y, t)∆blog|f(y)|dy.
By the CR heat kernel estimate, the same computations of (4.5) in Lemma 4.1 give us the inequality
(5.3) 1
V (Bx(r)) Z
Bx(r)
∆blog|f(y)|dy ≤Cw x, r2 .
Combing (5.3) and (4.12), we obtain
(5.4) 1
V (Bx(r)) Z
Bx(r)
∆blog|f(y)|dy≤C d r2
for any r >0 by (4.10). By the equality ordex(f) = 1
2n lim
r→0+
r2 V (Bex(r))
Z
Bex(r)
∆ loge |f|,
we know that for any ǫ >0,there is δ >0 such that (5.5)
2nordex(f)− r2 V (Bxe(r))
Z
Bex(r)
∆ loge |f|
< ǫ 6
for 0< r < δ <1. Now we separate the integration in (5.2) into two parts:
(5.6)
tw(x, t)
= 2t Z
M
H(x, y, t)∆blog|f(y)|dy
= 2t[
Z
Bx(δ)c
H(x, y, t)∆blog|f(y)|dy+ Z
Bx(δ)
H(x, y, t)∆blog|f(y)|dy].
So if we could show that (I) the first integration is close to zero and (II) the second one is close to ordx(f) ast →0+ in the last line of (5.6), then (5.1) holds.
(I). Now we’re going to show the first integration in (5.6) goes to zero for sufficiently small t as follows: By CR heat kernel estimate, we have, for t≤δ2,
(5.7)
2t Z
Bx(δ)c
H(x, y, t)∆blog|f(y)|dy
≤ 2t C
V(Bx(√t)) R+∞
δ exp
−Cr2t2 (
Z
∂Bx(r)
∆blog|f(s)|dσ(s))dr
≤ −V(BxC(√t))exp
−Cδ2t2 Z
Bx(δ)
∆blog|f(y)|dy +CtR+∞
δ [exp
−Cr2t2
r t
r
√t
2C
(V(B1x(r)) Z
Bx(r)
∆blog|f(y)|dy)]dr.
Here we use integration by part, (5.4) and the CR volume doubling property in the second inequality. From the CR volume doubling property and the inequality (5.4), we obtain
(5.8) lim
t→0+
C V Bx
√texp(−Cδ2 2t)
Z
Bx(δ)
∆blog|f(y)|dy= 0.
